# Definition:Lexicographic Order/Family

## Definition

Let $\struct {I, \preceq}$ be a well-ordered set.

For each $i \in I$, let $\struct {S_i, \preccurlyeq_i}$ be an ordered set.

Let $\ds D = \prod_{i \mathop \in I} S_i$ be the Cartesian product of the family $\family {\struct {S_i, \preccurlyeq_i} }_{i \mathop \in I}$ indexed by $I$.

Then the **lexicographic order** on $D$ is defined as:

- $\ds \struct {D, \preccurlyeq_D} := {\bigotimes_{i \mathop \in I} }^l \struct {S_i, \preccurlyeq_i}$

where $\preccurlyeq_D$ is defined as:

- $\forall u, v \in D: u \preccurlyeq_D v \iff \begin {cases} u = v \\ \map u i \preccurlyeq_i \map v i & \text {for the $\preceq$-smallest $i \in I$ such that $\map u i \ne \map v i$} \end {cases}$

## Also known as

**Lexicographic order** can also be referred to as the more unwieldy **lexicographical ordering**.

Some sources refer to it as **dictionary order**.

Some sources classify the **lexicographic order** as a variety of order product.

Hence the term **lexicographic product** can occasionally be seen.

The mathematical world is crying out for a less unwieldy term to use.

Some sources suggest **Lex**, but this has yet to filter through to general usage.

## Also see

- Results about
**the lexicographic order**can be found here.

## Sources

- 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations